9.8.3.1. Tables of Bravais lattices

The (3 + 1)-dimensional lattice is determined by the three-dimensional vectors a*, b*, c* and the modulation vector q. The former three vectors give by duality a, b, and c, the external components of lattice basis vectors, and the products , , and the corresponding internal components. Therefore, it is sufficient to give the arithmetic crystal class of the group and the components σj (σ1 = α, σ2 = β, and σ3 = γ) of the modulation vector q with respect to a conventional basis a*, b*, c*. The arithmetic crystal class is denoted by a modification of the symbol of the three-dimensional symmorphic space group of this class (see Chapter 1.4
) plus an indication for the row matrix σ (having entries ). In this way, one obtains the so-called one-line symbols used in Tables 9.8.3.1 and 9.8.3.2.

(a) (2 + 1)-Dimensional Bravais classes. The holohedral point group Ks is given in terms of its external and internal parts, KE, and KI, respectively. The reflections are given by ha* + kb* + mq where q is the modulation wavevector. If the rational part qr is not zero, there is a corresponding centring translation in three-dimensional space. The conventional basis (, , qi) given for the vector module M* is shown such that qr = 0. The basis vectors are given by components with respect to the conventional basis a*, b* of the lattice Λ* of main reflections.

No.

Symbol

KE

KI

q

Conventional basis

Centring

Oblique

1

2p(αβ)

2

(αβ)

(10), (01), (αβ)

Rectangular

2

mmp(0β)

mm

(0β)

(10), (01), (0β)

3

mmp(β)

mm

(β)

(0), (01), (0β)

0

4

mmc(0β)

mm

(0β)

(10), (01), (0β)

0

(b) (2 + 2)-Dimensional Bravais classes. The holohedral point group Ks is given in terms of its external and internal parts, KE and KI, respectively. The basis of the vector module M* contains two modulation wavevectors and the reflections are given by ha* + kb* + m1q1 + m2q2. If or are not zero, there are corresponding centring translations in four-dimensional space. The conventional basis (, , , ) for the vector module M* is chosen such that . The basis vectors are indicated by their components with respect to the conventional basis a*, b* of the lattice Λ* of main reflections.

(a) (3 + 1)-Dimensional Bravais classes for incommensurate structures. The holohedral point group Ks is given in terms of its external and internal parts, KE and KI, respectively. The reflections are given by ha* + kb* + lc* + mq, where q is the modulation wavevector. If the rational part qr is not zero, there is a corresponding centring translation in four-dimensional space. A conventional basis (, , , qi) for the vector module M* is then chosen such that qr = 0. The basis vectors are indicated by their components with respect to the conventional basis a*, b*, c* of the lattice Λ* of main reflections. The Bravais classes can also be found in Janssen (1969) and Brown et al. (1978). The notation of the Bravais classes there is here given in the columns Ref. a and Ref. b, respectively.

No.

Symbol

KE

KI

q

Conventional basis

Centring translation(s)

Ref. a

Ref. b

Triclinic

1

P(αβγ)

(αβγ)

(100), (010), (001), (αβγ)

I P

I/I

Monoclinic

2

2/mP(αβ0)

2/m

(αβ0)

(100), (010), (001), (αβ0)

II P

II/I

3

2/mP(αβ)

2/m

(αβ)

(100), (010), (00), (αβ0)

00

II I

II/II

4

2/mB(αβ0)

2/m

(αβ0)

(100), (010), (001), (αβ0)

II I

II/II

5

2/mP(00γ)

2/m

(00γ)

(100), (010), (001), (00γ)

III P

III/I

6

2/mP(0γ)

2/m

(0γ)

(00), (010), (001), (00γ)

00

III I

III/II

7

2/mB(00γ)

2/m

(00γ)

(100), (010), (001), (00γ)

00

III I

III/II

8

2/mB(0γ)

2/m

(0γ)

(100), (00), (001), (00γ)

00, 00

III G

III/III

Orthorhombic

9

mmmP(00γ)

mmm

(00γ)

(100), (010), (001), (00γ)

IV P

IV/I

10

mmmP(0γ)

mmm

(0γ)

(100), (00), (001), (00γ)

00

IV B

IV/III

11

mmmP(γ)

mmm

(γ)

(00), (00), (001), (00γ)

00, 00

IV F

IV/VI

12

mmmI(00γ)

mmm

(00γ)

(100), (010), (001), (00γ)

0

IV I

IV/IV

13

mmmC(00γ)

mmm

(00γ)

(100), (010), (001), (00γ)

00

IV C

IV/II

14

mmmC(10γ)

mmm

(10γ)

(100), (010), (001), (00γ)

0

IV I

IV/IV

15

mmmA(00γ)

mmm

(00γ)

(100), (010), (001), (00γ)

00

IV B

IV/III

16

mmmA(0γ)

mmm

(0γ)

(00), (010), (001), (00γ)

00, 00

IV G

IV/V

17

mmmF(00γ)

mmm

(00γ)

(100), (010), (001), (00γ)

00, 00

IV F

IV/VI

18

mmmF(10γ)

mmm

(10γ)

(100), (010), (001), (00γ)

0, 0

IV G

IV/V

Tetragonal

19

4/mmmP(00γ)

4/mmm

(00γ)

(100), (010), (001), (00γ)

VII P

VI/I

20

4/mmmP(γ)

4/mmm

(γ)

(0), (0), (001), (00γ)

0

VII I

VI/II

21

4/mmmI(00γ)

4/mmm

(00γ)

(100), (010), (001), (00γ)

0

VII I

VI/II

Trigonal

22

mR(00γ)

(00γ)

(100), (010), (001), (00γ)

0

VI P

VII/I

23

1mP(γ)

(γ)

(0), (0), (001), (00γ)

0

VI P

VII/I

Hexagonal

24

6/mmmP(00γ)

6/mmm

(00γ)

(100), (010), (001), (00γ)

V P

VII/II

(b) (3 + 1)-Dimensional Bravais classes for commensurate structures. The holohedral point group Ks is given in terms of its external and internal parts, KE and KI, respectively. The reflections are given by ha* + kb* + lc* + mq, where q is the modulation wavevector. Here q is a commensurate vector having rational components with respect to a*, b*, c*. The rank of the vector module M* is three. Therefore, there are three basis vectors for M*. They are given by their components with respect to the conventional basis a*, b*, c* of the lattice of main reflections. If they do not coincide with the primitive basis vectors of the lattice Λ* of main reflections, there is a centring in four-dimensional space. The notation of the Bravais classes in Janssen (1969) is here given in the column Ref. a. Notice that for a commensurate one-dimensional modulation cubic symmetry is also possible.

No.

Symbol

KE

KI

q

Conventional basis

Centring translation(s)

Ref. a

Triclinic

1

P(000)

(000)

(100), (010), (001)

II P

2

P()

()

(), (),

II I

Monoclinic

3

2/mP(000)

2/m1

(000)

(100), (010), (001)

IV P

4

2/mP(0)

2/m1

(0)

(0), (0), (001)

0

IV B

5

2/mP(00)

2/m1

(00)

(100), (010), (00)

00

IV C

6

2/mP()

2/m1

()

(0), (0), (00)

0, 00

IV F

7

2/mB(000)

2/m1

(000)

(100), (010), (001)

00

IV B

8

2/mB(100)

2/m1

(100)

(100), (010), (001)

0

IV I

9

2/mB(00)

2/m1

(00)

(100), (00), (001)

00, 00

IV G

Orthorhombic

10

mmmP(000)

mmm1

(000)

(100), (010), (001)

VIII P

11

mmmP(00)

mmm1

(00)

(100), (010), (00)

00

VIII A

12

mmmP(0)

mmm1

(0)

(100), (00), (00)

00, 00

VIII F

13

mmmP()

mmm1

()

(00), (00), (00)

00, 00, 00

VIII S

14

mmmI(000)

mmm1

(000)

(100), (010), (001)

0

VIII E

15

mmmI(111)

mmm1

(111)

(100), (010), (001)

VIII I

16

mmmI()

mmm

()

(00), (00), (00)

, ,

VIII K

17

mmmF(000)

mmm1

(000)

(100), (010), (001)

00, 00

VIII F

18

mmmF(001)

mmm1

(001)

(100), (010), (001)

00, 0

VIII H

19

mmmC(000)

mmm1

(000)

(100), (010), (001)

00

VIII A

20

mmmC(100)

mmm1

(100)

(100), (010), (001)

0

VIII E

21

mmmC(00)

mmm1

(00)

(100), (010), (00)

00, 00

VIII G

22

mmmC(10)

mmm1

(10)

(100), (010), (00)

0, 0

VIII H

Tetragonal

23

4/mmmP(000)

4/mmm1

(000)

(100), (010), (001)

XII P

24

4/mmmP(00)

4/mmm1

(00)

(100), (010), (00)

00

XII A

25

4/mmmP(0)

4/mmm1

(0)

(0), (0), (001)

0

XII E

26

4/mmmP()

4/mmm1

()

(0), (0), (00)

0, 00

XII H

27

4/mmmI(000)

4/mmm1

(000)

(100), (010), (001)

0

XII E

28

4/mmmI(111)

4/mmm1

(111)

(100), (010), (001)

XII I

29

4/mmmI()

4/mmm

()

(100), (010), (001)

XII N

Trigonal

30

m1R(000)

(000)

(100), (010), (001)

0

X R

31

m1R(00)

(00)

(100), (010), (00)

00,

X RI

Hexagonal

32

6/mmmP(000)

6/mmm1

(000)

(100), (010), (001)

X P

33

6/mmmP(00)

6/mmm1

(00)

(100), (010), (00)

00

X A

34

6/mmmP(0)

6/mmm

(0)

(0), (0), (001)

0

X R

35

6/mmmP()

6/mmm

()

(0), (0), (00)

0, 00

X RI

Cubic

36

m3mP(000)

(000)

(100), (010), (001)

XIV P

37

m3mP()

()

(00), (00), (00)

00, 00, 00

XIV S

38

m3mI(000)

(000)

(100), (010), (001)

0

XIV V

39

m3mI(111)

(111)

(100), (010), (001)

XIV I

40

m3mI()

()

(00), (00), (00)

, ,

XIV K

41

m3mF(000)

(000)

(100), (010), (001)

00, 00

XIV F

As an example, the symbol denotes a Bravais class for which the main reflections belong to a B-centred monoclinic lattice (unique axis c) and the satellite positions are generated by the point-group transforms of . Then the matrix σ becomes . It has as irrational part and as rational part . The external part of the (3 + 1)-dimensional point group of the Bravais lattice is 2/m. By use of the relation [cf. (9.8.2.4)] we see that the operations 2 and m are associated with the internal space transformations = 1 and = −1, respectively. This is denoted by the one-line symbol for the (3 + 1)-dimensional point group of the Bravais lattice. In direct space, the symmetry operation {R, (R)} is represented by the matrix Γ(R) which transforms the components , of a vector to: The operations (2, 1) and are represented by the matrices: The 3 × 3 part of each matrix is obtained by considering the action of R on the external part v of . The 1 × 1 part is the value of the associated with R and the remaining part follows from the relation

Bravais classes can be denoted in an alternative way by two-line symbols. In the two-line symbol, the Bravais class is given by specifying the arithmetic crystal class of the external symmetry by the symbol of its symmorphic space group, the associated elements by putting their symbol under the corresponding symbols of , and by the rational part indicated by a prefix. In the following table, this prefix is given for the components of that play a role in the classification. Note that the integers appearing here are not equivalent to zero because they express components with respect to a conventional lattice basis (and not a primitive one). For the Bravais class mentioned above, the two-line symbol is . This symbol has the advantage that the internal transformation (the value of ) is explicitly given for the corresponding generators. It has, however, certain typographical drawbacks. It is rare for the printer to put the symbol together in the correct manner: .

In Tables 9.8.3.1 and 9.8.3.2 the symbols for the (2 + d)- and (3 + 1)-dimensional Bravais classes are given in the one-line form. It is, however, easy to derive from each one-line symbol the corresponding two-line symbol because the bottom line for the two-line symbol appears in the tables as the internal part of the point-group symbol.

The number of symbols in the bottom line of the two-line symbol should be equal to that of the generators given in the top line. A symbol `1' is used in the bottom line if the corresponding is the unit transformation. If necessary, a mirror perpendicular to a crystal axis is indicated by and one that is not by . This situation only occurs for . So the (2 + 2)-dimensional class is actually and is different from the class . In a one-line symbol, their difference is apparent, the first being 4mp(α0), whereas the second is 4mp(αα).

The four-dimensional point group Ks has external part KE, which belongs to a three-dimensional system. Depending on the Bravais class of the four-dimensional lattice left invariant by Ks, this point group gives rise to an integral 4 × 4 matrix group Γ(K) which belongs to one of the arithmetic crystal classes given in the last column.

System

Point group

External Bravais class

Arithmetic crystal class(es)

KE

Ks

Triclinic

1

(1, 1)

P

1P(αβγ)

()

P

P(αβγ)

Monoclinic

2

()

2/mP

2P(αβ0), 2P(αβ)

2/mB

2B(αβ0)

(2, 1)

2/mP

2P(00γ), 2P(0γ)

2/mB

2B(00γ), 2B(0γ)

m

(m, 1)

2/mP

mP(αβ0), mP(αβ)

2/mB

mB(αβ0)

(m, )

2/mP

mP(00γ), mP(0γ)

2/mB

mB(00γ), mB(0γ)

2/m

(2/m, )

2/mP

2/mP(αβ0), 2/mP(αβ)

2/mB

2/mB(αβ0)

(2/m, )

2/mP

2/mP(00γ), 2/mP(0γ)

2/mB

2/mB(00γ), 2/mB(0γ)

Orthorhombic

222

(222, )

mmmP

222P(00γ), 222P(0γ), 222P(γ)

mmmI

222I(00γ)

mmmF

222F(00γ), 222F(10γ)

mmmC

222C(00γ), 222C(10γ)

(222, )

mmmC

222C(α00), 222C(α0)

mm2

(mm2, 111)

mmmP

mm2P(00γ), mm2P(0γ), mm2P(γ)

mmmI

mm2I(00γ)

mmmF

mm2F(00γ), mm2F(10γ)

mmmC

mm2C(00γ), mm2C(10γ)

(2mm, 111)

mmmC

2mmC(α00), 2mmC(α0)

(2mm, )

mmmP

2mmP(00γ), 2mmP(0γ), 2mmP(γ)

mmmI

2mmI(00γ)

mmmF

2mmF(00γ), 2mmF(10γ)

mmmC

2mmC(00γ), 2mmC(10γ)

(mm2, )

mmmC

mm2C(α00), mm2C(α0)

(m2m, )

mmmP

m2mP(0γ)

(m2m, )

mmmC

m2mC(α00), m2mC(α0)

mmm

(mmm, )

mmmP

mmmP(00γ), mmmP(0γ), mmmP(γ)

mmmI

mmmI(00γ)

mmmF

mmmF(00γ), mmmF(10γ)

mmmC

mmmC(00γ), mmmC(10γ)

(mmm, )

mmmC

mmmC(α00), mmmC(α0)

Tetragonal

4

(4, 1)

4/mmmP

4P(00γ), 4P(γ)

4/mmmI

4I(00γ)

(, )

4/mmmP

P(00γ), P(γ)

4/mmmI

I(00γ)

4/m

(4/m, )

4/mmmP

4/mP(00γ), 4/mP(γ)

4/mmmI

4/mI(00γ)

422

(422, )

4/mmmP

422P(00γ), 422P(γ)

4/mmmI

422I(00γ)

4mm

(4mm, 111)

4/mmmP

4mmP(00γ), 4mmP(γ)

4/mmmI

4mmI(00γ)

2m

(2m, )

4/mmmP

2mP(00γ), 2mP(γ)

4/mmmI

2mI(00γ)

m2

(m2, )

4/mmmP

m2P(00γ), m2P(γ)

4/mmmI

m2I(00γ)

4/mmm

(4/mmm, )

4/mmmP

4/mmmP(00γ), 4/mmmP(γ)

4/mmmI

4/mmmI(00γ)

Trigonal

3

(3, 1)

mR

3R(00γ)

6/mmmP

3P(00γ), 3P(γ)

(, )

mR

R(00γ)

6/mmmP

P(00γ), P(γ)

32

(32, )

mR

32R(00γ)

6/mmmP

312P(00γ), 312P(γ), 321P(00γ)

3m

(3m, 11)

mR

3mR(00γ)

6/mmmP

3m1P(00γ), 31mP(00γ), 31mP(γ)

m

(, )

mR

mR(00γ)

6/mmmP

1mP(00γ), 1mP(γ), m1P(00γ)

Hexagonal

6

(6, 1)

6/mmmP

6P(00γ)

(, )

6/mmmP

P(00γ)

6/m

(6/m, )

6/mmmP

6/mP(00γ)

622

(622, )

6/mmmP

622P(00γ)

6mm

(6mm, 111)

6/mmmP

6mmP(00γ)

m2

(, )

6/mmmP

m2P(00γ)

(, )

6/mmmP

2mP(00γ)

6/mmm

(6/mmm, )

6/mmmP

6/mmmP(00γ)

The symbols for geometric crystal classes indicate the pairs of the generators of the point group. This is done by giving the crystal class for the point group and the symbols for the corresponding elements of . So, for example, the geometric crystal class belonging to the holohedral point group of the Bravais class , mentioned above, is .

The notation for the arithmetic crystal classes is similar to that for the Bravais classes. In the tables, their one-line symbols are given. They consist of the (modified) symbol of the three-dimensional symmorphic space group and, in parentheses, the appropriate components of the modulation wavevector. The three arithmetic crystal classes implying a lattice belonging to the Bravais class are , , and . The corresponding geometric crystal classes are , , and .

9.8.3.3. Tables of superspace groups

9.8.3.3.1. Symmetry elements

The transformations belonging to a (3 + 1)-dimensional superspace group consist of a point-group transformation given by the integral matrix Γ(R) and of the associated translation. So the superspace group is determined by the arithmetic crystal class of its point group and the corresponding translational components. The symbol for the arithmetic crystal class has been discussed in Subsection 9.8.3.2. Given a point-group transformation , the associated translation is determined up to a lattice translation. As in three dimensions, the translational part generally depends on the choice of origin. To avoid this arbitrariness, one decomposes that translation into a component (called intrinsic) independent of the origin, and a remainder. The (3 + 1)-dimensional translation associated with the point-group transformation is given by Its origin-invariant part is given by where n is now the order of the point-group transformation R so that is the identity. As customary also in three-dimensional crystallography, one indicates in the space-group symbol the invariant components . Notice that this means that there is an origin for in (3 + 1)-dimensional superspace such that the translation associated with has these components. This origin, however, may not be the same for different transformations , as is known in three-dimensional crystallography.

Written in components, the non-primitive translation associated with the point-group element is , where can be written as . In accordance with (9.8.1.12), δ is defined as . The origin-invariant part of is where The internal transformation = (R) = is either +1 or −1. When = −1 it follows from (9.8.3.6) that . For , one has . Because in that case it follows that

For of order n, is the identity and the associated translation is a lattice translation. The ensuing values for τ are or (modulo integers). This remains true also in the case of a centred basis. The symbol of the (3 + 1)-dimensional space-group element is determined by the invariant part of its three-dimensional translation and τ. Again, that information can be given in terms of either a one-line or a two-line symbol.

In the one-line symbol, one finds: the symbol according to International Tables for Crystallography, Volume A, for the space group generated by the elements {R|v}, in parentheses the components of the modulation vector q followed by the values of τ, one for each generator appearing in the three-dimensional space-group symbol. A letter symbolizes the value of τ according to As an example, consider the superspace group The external components of the elements of this group form the three-dimensional space group . The modulation wavevector is αa* + βb* with respect to a conventional basis of the monoclinic lattice with unique axis c. Therefore, the point group is . The point-group element has associated a non-primitive translation with invariant part and the point-group generator (m, 1) one with .

In the two-line symbol, one finds in the upper line the symbol for the three-dimensional space group, in the bottom line the value of τ for the case = +1 and the symbol `' when = −1. The rational part of q is indicated by means of the appropriate prefix. In the case considered, qr = 000. So the prefix is P and the same superspace group is denoted in a two-line symbol as In Table 9.8.3.5, the (3 + 1)-dimensional space groups are given by one-line symbols. These are so-called short symbols. Sometimes, a full symbol is required. Then, for the example given above one has and , respectively. Note that in the short one-line symbol for τ = 0 superspace groups (where the non-primitive translations can be transformed to zero by a choice of the origin) the zeros for the translational part are omitted. Not so, of course, in the full symbol. For example, short symbol P21/m(αβ0) and full symbol P1121/m(αβ0)0000. Table 9.8.3.5 is an adapted version of the tables given by de Wolff, Janssen & Janner (1981) and corrected by Yamamoto, Janssen, Janner & de Wolff (1985).

9.8.3.3.2. Reflection conditions

The indexing of diffraction vectors is a matter of choice of basis. When the basis chosen is not a primitive one, the indices have to satisfy certain conditions known as centring conditions. This holds for the main reflections (centring in ordinary space) as well as for satellites (centring in superspace). These centring conditions for reflections have been discussed in Subsection 9.8.2.1.

In addition to these general reflection conditions, there may be special reflection conditions related to the existence of non-primitive translations in the (3 + 1)-dimensional space group, just as is the case for glide planes and screw axes in three dimensions.

Special reflection conditions can be derived from transformation properties of the structure factor under symmetry operations. Transforming the geometric structure factor by an element , one obtains Therefore, if RH = H, the corresponding structure factor vanishes unless is an integer.

The form of such a reflection condition in terms of allowed or forbidden sets of indices depends on the basis chosen. When a lattice basis is chosen, one has Then the reflection condition becomes

In terms of external and internal shift components, the reflection condition can be written as With and , (9.8.3.14) gives For and , (9.8.3.15) takes the form (9.8.3.13): When the modulation wavevector has a rational part, one can choose another basis (Subsection 9.8.2.1) such that K′ = K + mqr has integer coefficients: Then, (9.8.3.15) with becomes and (9.8.3.16) transforms into in which , and are the components of v with respect to the basis , and .

As an example, consider a (3 + 1)-dimensional space-group transformation with R a mirror perpendicular to the x axis, , and with b orthogonal to a. The modulation wavevector is supposed to be . Then . The vectors H left invariant by R satisfy the relation 2h + m = 0. For such a vector, the reflection condition becomes For the basis , , c*, the rational part of the wavevector vanishes. The indices with respect to this basis are H = h + k + m, K = h − k, L = l and m. The condition now becomes Of course, both calculations give the same result: k + m = 2n for h, k, l, −2h and H − K + m = 4n for H, −H, L, m.

The special reflection conditions for the elements occurring in (3 + 1)-dimensional space groups are given in Table 9.8.3.5.

9.8.3.4. Guide to the use of the tables

In the tables, Bravais classes, point groups, and space groups are given for three-dimensional incommensurate modulated crystals with a modulation of dimension one and for two-dimensional crystals (e.g. surfaces) with one- and two-dimensional modulation (Janssen, Janner & de Wolff, 1980). In the following, we discuss briefly the information given. Examples of their use can be found in Subsection 9.8.3.5.

To determine the symmetry of the modulated phase, one first determines its average structure, which is obtained from the main reflections. Since this structure has three-dimensional space-group symmetry, this analysis is performed in the usual way.

The diffraction pattern of the three-dimensional modulated phase can be indexed by 3 + 1 integers. The Bravais class is determined by the symmetry of the vector module spanned by the 3 + 1 basis vectors. The crystallographic system of the pattern is equal to or lower than that implied by the main reflections. One chooses a conventional basis (qr = 0) for the vector module, and finds the Bravais class from the general reflection conditions using Table 9.8.3.6. The relation between indices hklm with respect to the basis a*, b*, c*, and q and HKLm with respect to the conventional basis , , , and is also given there.

Table 9.8.3.2(a) gives the number labelling the (3 + 1)-dimensional Bravais class, its symbol, its external and internal point group, and the modulation wavevector. Moreover, the superspace conventional basis (for which the rational part qr vanishes) and the corresponding (3 + 1)-dimensional centring are given. Because the four-dimensional lattices belong to Euclidean Bravais classes, the corresponding class is also given in the notation of Janssen (1969) and Brown et al. (1978).

The point group of the modulated structure is a subgroup of the holohedry of its lattice Λ. In Table 9.8.3.3, for each system the (3 + 1)-dimensional point groups are given. Each system contains one or more Bravais classes. Each geometric crystal class contains one or more arithmetic crystal classes. The (3 + 1)-dimensional arithmetic classes belonging to a given geometric crystal class are also listed in Table 9.8.3.3.

Starting from the space group of the average structure, one can determine the (3 + 1)-dimensional superspace group. In Table 9.8.3.5, the full list of these (3 + 1)-dimensional superspace groups is given for the incommensurate case and are ordered according to their basic space group. They have a number n.m where n is the number of the basic space group one finds in International Tables for Crystallography, Volume A. The various (3 + 1)-dimensional superspace groups for each basic group are distinguished by the number m. Furthermore, the symbol of the basic space group, the point group, and the symbol for the corresponding superspace group are given. In the last column, the special reflection conditions are listed for typical symmetry elements. These may help in the structure analysis. The (2 + d)-dimensional superspace groups, relevant for modulated surface structures, are given in Table 9.8.3.4.

(a) (2 + 1)-Dimensional superspace groups. The number labelling the superspace group is denoted by n.m, where n is the number attached to the two-dimensional basic space group and m numbers the various superspace groups having the same basic space group. The symbol of the basic space group, the symbol for the three-dimensional point group, the number of the three-dimensional Bravais class to which the superspace group belongs (Table 9.8.3.1a) and the superspace-group symbol are also given.

No.

Basic space group

Point group Ks

Bravais class No.

Group symbol

Oblique

1.1

p1

(1, 1)

1

p1(αβ)

2.1

p2

(2, )

1

p2(αβ)

Rectangular

3.1

pm

(m, 1)

2

pm1(0β)

3.2

2

pm1(0β)s0

3.3

3

pm1(β)

3.4

(m, )

2

p1m(0β)

3.5

3

p1m(β)

4.1

pg

(m, 1)

2

pg1(0β)

4.2

3

pg1(β)

4.3

(m, )

2

p1g(0β)

5.1

cm

(m, 1)

4

cm1(0β)

5.2

4

cm1(0β)s0

5.3

(m, )

4

c1m(0β)

6.1

pmm

(mm, )

2

pmm(0β)

6.2

2

pmm(0β)s0

6.3

3

pmm(β)

7.1

pmg

(mm, )

2

pmg(0β)

7.2

2

pgm(0β)

7.3

3

pgm(β)

8.1

pgg

(mm, )

2

pgg(0β)

9.1

cmm

(mm, )

4

cmm(0β)

9.2

4

cmm(0β)s0

(b) (2 + 2)-Dimensional superspace groups. The number labelling the superspace group is denoted by n.m, where n is the number attached to the two-dimensional basic space group and m numbers the various superspace groups having the same basic space group. The symbol of the basic space group, the symbol for the four-dimensional point group, the number of the four-dimensional Bravais class to which the superspace group belongs (Table 9.8.3.1b) and the superspace-group symbol are also given.

No.

Basic space group

Point group Ks

Bravais class No.

Group symbol

Oblique

1.1

p1

(1, 1)

1

p1(αβ, λμ)

2.1

p2

(2, 2)

1

p2(αβ, λμ)

Rectangular

3.1

pm

(m, 1)

2

pm1(0β, 0μ)

3.2

2

pm1(0β, 0μ)s0, 0

3.3

3

pm1(β, 0μ)

3.4

3

pm1(β, 0μ)s0, 0

3.5

(m, 2)

2

p1m(0β, 0μ)

3.6

3

p1m(β, 0μ)

3.7

(m, m)

4

pm1(α0, 0μ)

3.8

4

pm1(α0, 0μ)0s, 0

3.9

5

pm1(α, 0μ)

3.10

5

pm1(α, 0μ)0s, 0

3.11

5

p1m(α, 0μ)

3.12

5

p1m(α, 0μ)0, s0

3.13

6

pm1(α, μ)

3.14

7

pm1(αβ)

4.1

pg

(m, 1)

2

pg1(0β, 0μ)

4.2

3

pg1(β, 0μ)

4.3

(m, 2)

2

p1g(0β, 0μ)

4.4

(m, m)

4

pg1(α0, 0μ)

4.5

7

pg1(α, μ)

5.1

cm

(m, 1)

8

cm1(0β, 0μ)

5.2

8

cm1(0β, 0μ)s0, 0

5.3

(m, 2)

8

c1m(0β, 0μ)

5.4

(m, m)

9

cm1(α0, 0μ)

5.5

9

cm1(α0, 0μ)0s, 0

5.6

10

cm(αβ)

6.1

pmm

(mm, 12)

2

pmm(0β, 0μ)

6.2

2

pmm(0β, 0μ)s0, 0

6.3

3

pmm(β, 0μ)

6.4

3

pmm(β, 0μ)s0, 0

6.5

(mm, mm)

4

pmm(α0, 0μ)

6.6

4

pmm(α0, 0μ)0s, 0

6.7

4

pmm(α0, 0μ)0s, s0

6.8

5

pmm(α, 0μ)

6.9

5

pmm(α, 0μ)0s, 0

6.10

6

pmm(α, μ)

6.11

7

pmm(αβ)

7.1

pmg

(mm, 12)

2

pmg(0β, 0μ)

7.2

2

pmg(0β, 0μ)0s, 0

7.3

2

pgm(0β, 0μ)

7.4

3

pgm(β, 0μ)

7.5

(mm, mm)

4

pgm(α0, 0μ)

7.6

4

pgm(α0, 0μ)0, s0

7.7

5

pmg(α, 0μ)

7.8

5

pmg(α, 0μ)0s, 0

7.9

7

pgm(αβ)

8.1

pgg

(mm, 12)

2

pgg(0β, 0μ)

8.2

(mm, mm)

4

pgg(α0, 0μ)

8.3

7

pgg(αβ)

9.1

cmm

(mm, 12)

8

cmm(0β, 0μ)

9.2

8

cmm(0β, 0μ)0s, 0

9.3

(mm, mm)

9

cmm(α0, 0μ)

9.4

9

cmm(α0, 0μ)0s, 0

9.5

9

cmm(α0, 0μ)0s, s0

9.6

10

cmm(αβ)

Tetragonal

10.1

p4

(4, 4)

11

p4(αβ)

11.1

p4m

(4m, 4m)

12

p4m(α0)

11.2

12

p4m(α0)0, 0s

11.3

13

p4m(α)

11.4

(, )

14

p4m(αα)

11.5

14

p4m(αα)0, 0s

12.1

p4g

(4m, 4m)

12

p4g(α0)

12.2

12

p4g(α0), 0s

12.3

(, )

14

p4g(αα)

12.4

14

p4g(αα)0, 0s

Hexagonal

13.1

p3

(3, 3)

15

p3(αβ)

14.1

p3m1

(3m, 3m)

16

p3m1(α0)

14.2

(, )

17

p3m1(αα)

15.1

p31m

(3m, 3m)

16

p31m(α0)

16.1

p6

(6, 6)

15

p6(αβ)

17.1

p6m

(6m, 6m)

16

p6m(α0)

17.2

(, )

17

p6m(αα)

9.8.3.5. Examples

Na2CO3 has a phase transition at about 753 K from the hexagonal to the monoclinic phase. At about 633 K, one vibration mode becomes unstable and below the transition temperature Ti = 633 K there is a modulated γ-phase (de Wolff & Tuinstra, 1986). At low temperature (128 K), a transition to a commensurate phase has been reported.

The main reflections in the modulated phase belong to a monoclinic lattice, and the satellites to a modulation with wavevector q = αa* + γc*, b axis unique. The dimension of the modulation is one. The main reflections satisfy the condition Therefore, the lattice of the average structure is C-centred monoclinic. For the satellites, the same general condition holds (hklm, h + k = even). From Table 9.8.3.6, one sees after a change of axes that the Bravais class of the modulated structure is Table 9.8.3.2(a) shows that the point group of the vector module is . The point group of the modulated structure is equal to or a subgroup of this one.

The space group of the average structure determined from the main reflections is C2/m (No. 12 in International Tables for Crystallography, Volume A). The superspace group may then be determined from the special reflection condition using Table 9.8.3.5. There are five superspace groups with basic group No. 12. Among them there are two in Bravais class 4. The reflection condition mentioned leads to the group In principle, the superspace group could be a subgroup of this, but, since the transition normal–incommensurate is of second order, Landau theory predicts that the basic space group is the symmetry group of the unmodulated monoclinic phase, which is .

The number labelling the superspace group is denoted by n.m, where n is the number attached to the three-dimensional basic space group and m numbers the various superspace groups having the same basic space group. The symbol of the basic space group, the symbol for the four-dimensional point group Ks, the number of the four-dimensional Bravais class to which the superspace group belongs (Table 9.8.3.2a), and the superspace-group symbol are also given. The superspace-group symbol is indicated in the short notation, i.e. for the basic group one uses the short symbol from International Tables for Crystallography, Volume A, and then the values of τ are given for each of the generators in this symbol, unless all these values are zero. Then, instead of writing a number of zeros, one omits them all. Finally, the special reflection conditions due to non-primitive translations are given, for hklm if qr = 0 and for HKLm otherwise. Recall the HKLm are the indices with respect to a conventional basis as in Table 9.8.3.2(a). The reflection conditions due to centring translations are given in Table 9.8.3.6.